Inference By Enumeration
By: Chris McCabe
CS5300 - Artificial Intelligence

The derivation of Bayes formula:
P(A | B) = P(A & B) / P(B)

Using some algebra, this equation can be rewritten as:
P(A & B) = P(A | B) P(B)

The same right-hand value can also be computed using A as the conditioning variable:
P(A & B) = P(B | A) P(A)

Given this equivalence, you can write:
P(A | B) P(B) = P(B | A) P(A)

Simplifying, you arrive at Bayes theorem:
P(A | B) = P(B | A) P(A) / P(B)

Now that we have Bayes formula we can begin to calculate the inference by enumeration.

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p(b | a):
First thing we need to do is rewrite this equation using bayes formula:
= p(b, a) / [p(b, a) + p(枴, a)]

Next we need to calculate the probablity of p(b,a)
p(b, a) = p(b)[p(e)p(a | b, e) + p(枸)p(a | b, 枸)]
All of these values we can pull directly from the tables given in the alarm network.
= .001(.002*.95 + .998*.94) = 0.00094

Now, we need to caclulate the probability of p(枴, a):
p(枴, a)= p(枴)[p(e)p(a | 枴, e) + p(枸)p(a | 枴, 枸)]
Once again we can pull these values from the table.
= .999(.002*.29+.998*.001) = 0.00158

Now, after calculating these probabilities we can finally calculate the final result:
= 0.00094 / (0.00094 + 0.00158) = 0.373

To help solidify this we will run through one more example using our alarm network:

p(b | e, a) = p(b, a, e) / [p(b, a, e) + p(枴, a, e)]
p(b, a, e) = p(b)p(e)p(a | b, e)
= .001(.002*.95 ) = 0.0000019

p(枴, a, e)= p(枴)p(e)p(a | 枴, e)
= .999(.002*.29) = 0.000579

= 0.0000019/ (0.0000019+ 0.000579) = 0.003

That is how we compute inference by enumeration.